We address the image formation of a dynamic object from projections by formulating it as a state estimation problem. The problem is solved with the ensemble Kalman filter (EnKF), a Monte Carlo algorithm that is computationally tractable when the state dimension is large. In this paper, we first rigorously address the convergence of the EnKF. Then, the effectiveness of the EnKF is demonstrated in a numerical experiment where a highly variable object is reconstructed from its projections, an imaging modality not yet explored with the EnKF. The results show that the EnKF can yield estimates of almost equal quality as the optimal Kalman filter but at a fraction of the computational effort. Further experiments explore the rate of convergence of the EnKF, its performance relative to an idealized particle filter, and implications of modeling the system dynamics as a random walk.

This paper formally addresses the asymptotic convergence of the ensemble Kalman filter (EnKF), a state estimation procedure that, when combined with a technique called localization, provides computationally tractable solutions to large-dimensional state estimation problems. The proof presented in this paper shows that the estimates given by the EnKF converge to the optimal estimates given by the Kalman filter (KF) and provides a formal justification for the use of the EnKF in dynamic remote sensing image formation. The implications of the proof are twofold: it shows that the EnKF converges to a well-defined limit and provides a formal argument that the EnKF is in fact a Monte Carlo algorithm that converges to the KF.

We address the reconstruction of a physically evolving unknown from tomographic measurements by formulating it as a state estimation problem. The approach presented in this paper is the localized ensemble Kalman filter (LEnKF); a Monte Carlo state estimation procedure that is computationally tractable when the state dimension is large. We establish the conditions under which the LEnKF is equivalent to the Gaussian particle filter. The performance of the LEnKF is evaluated in a numerical example and is shown to give state estimates of almost equal quality as the optimal Kalman filter but at a 95% reduction in computation.